On a conjecture due to Kanade related to Nahm sums

Abstract

Kanade explored the construction of modular companions to $q$-series identities, using the asymptotics of Nahm sums, and Mizuno [Ramanujan J. 66 (2025), Article 62, 31 pp.] recently obtained a generalization of Kanade's asymptotic formula for symmetrizable Nahm sums. A related conjecture from Kanade concerning the dilogarithm function and related to the work of Kurşungöz on Andrews--Gordon-type series [Ann. Comb. 23 (2019), 835--888] has remained open. In this paper, we prove Kanade's conjecture, through an application of dilogarithm identities due to Kirillov together with a dilogarithm ladder due to Loxton and Lewin. Inspired by Kanade's result, we extend this to prove two new related dilogarithm identities.

12 pages